 
Summary: Economical elimination of cycles in the torus
Dedicated to Tom Trotter, for his 65th birthday
Noga Alon
Abstract
Let m > 2 be an integer, let C2m denote the cycle of length 2m on the set of vertices
[m, m) = {m, m + 1, . . . , m  2, m  1}, and let G = G(m, d) denote the graph on the
set of vertices [m, m)d
, in which two vertices are adjacent iff they are adjacent in C2m in one
coordinate, and equal in all others. This graph can be viewed as the graph of the ddimensional
torus. We prove that one can delete a fraction of at most O(log d
m ) of the vertices of G so that
no topologically nontrivial cycles remain. This is tight up to the log d factor and improves
earlier estimates by various researchers.
1 Introduction
Let G = G(m, d) denote the graph on the set of vertices [m, m)d in which two vertices are
adjacent iff they are equal in all coordinates but one, in which they are adjacent in the cycle C2m.
This graph can be viewed as the graph of the ddimensional torus. A cycle in it is called nontrivial
if it wraps around the torus. In particular, the projection of each such cycle along at least one of
the coordinates contains all vertices of C2m. A (vertex) spine is a set of vertices that intersects
every nontrivial cycle. It is easy to see that there is a spine containing a fraction of O(d/m) of
